JOURNAL OF ALGEBRA 102, 444462 (1986)

A Homology Vanishing Theorem for Kac-Moody Algebras

with Coefficients in the Category 0

SHRAWAN KUMAR*

Mathematical Sciences Research Institute, Berkeley, California, and School of Mathematics, Tala Institute of Fundamental

Research, Homi Bhabha Road, Colaba, Bombay 400005, India

Communicated by J. Tits

Received February 16, 1984

INTRODUCTION

In this paper, we study the homology of a Kac-Moody Lie-algebra, associated to a symmetrizable generalized Cartan matrix, with coefficients in a module M in the category 0. We show that all the homology groups vanish if the Casimir operator acts as an automorphism on M. In fact we prove a slightly more general theorem (Theorem (1.2)). This is one of the main theorems of this paper.

We further study the homology of g with coefficients in arbitrary Verma modules and also integrable highest weight modules. WC prove (Proposition (1.5)) that Hi(g, M(A)) =0 for all i>O and for all the Verma modules M(A) with highest weight ,? E h* such that i is not equal to wp - p for any w E W, whereas Hi(g, M( wp - p)) w /1 ip ‘(“j(h). This result in the case when g is a finite dimensional semi-simple Lie-algebra is due to Williams.

If L(1) is an integrable highest weight module with i #O (2 is automatically dominant integral) then we have, using BGG resolution and Proposition (1.5), that Hi(g, L(;i)) = 0 for all ia 0. This is the content of our Theorem (1.7). In the case when g is finite dimensional, the integrable highest weight modules are precisely the finite dimensional irreducible modules. In this case, this theorem is well known and is due to Whitehead. As in the finite dimensional case, the situation when ,J = 0 (so that L(I1) is the trivial one dimensional module) is drastically different. We have recently proved [Ku; Theorem 1.61 that the homology of the commutator

* Research supported in part by NSF Grant 8120790

444 OO21-8693/86 $3.00 Copy&t 0 1986 by Academic Press, Inc AII rights of reproduction m any form reserved

HOMOLOGY OF KAC-MOODY ALGEBRAS 445

subalgebra [g, g] with trivial coefficients is isomorphic with the singular homology of the associated algebraic group G. Kac-Peterson also claim to prove this result [K6]. A formula for the Poincare series of H,(G) is given in [K6] in terms of the Weyl group. Further, H,(g, k) can be easily com- puted from H,( [g, g], k). See Proposition (1.9).

Finally, it seems like a reasonable conjecture to make that Hi(g, V(n)) = 0 for all the highest weight modules V(1) with highest weight 1 E h* such that A is not equal to wp - p for any w E W.

The main difficulty in carrying over the proof of Theorem (1.2) from the finite dimensional semi-simple Lie-algebras to the KaccMoody Lie- algebras lies in the fact that in the case of infinite dimensional Kac-Moody Lie-algebras, the Casimir operator does not operate on all the g-modules (e.g., it does not operate on U(g)-free modules). In particular the Casimir operator is not represented by a central element in U(g). So the usual (almost trivial) homological algebra argument, which works for finite dimensional semi-simple Lie-algebras g due to the presence of Casimir as a central element in U(g), does not carry over (without a change) to the general Kac-Moody Lie-algebras. At this point, it may be of some interest to note that Chari and Ilangovan have shown [CI] that the centre of U(g), in general, is too small to be of much interest.

To remedy this, we make use of an associative algebra o(g), given by Chari and Ilangovan [CI], which contains U(g) as a subalgebra. The Casimir lies in o(g) as a central element and moreover all the left U(g)- modules M in the category 0 admit an o(g)-module structure extending the U(g)-module structure on it. We prove the homology vanishing theorem by showing tha,t (Lemma (2.3)) HJg, M) (= Toru(s)(k, M)) z Tor,!‘(a)(k, M) for all left U(g)-modules M. The main ingredient in the proof of this isomorphism is the following proposition (Proposition (2.1)):

Hi(9, Q9N = 0 for all i3 1.

The proof of this proposition occupies Section (2). Once we have this proposition at our disposal, the usual homological algebra (see, e.g., [G]) and a lemma on the structure of I?(g) (Lemma (2.2)) take care of the rest. I feel that the vanishing of Hi(g, o(g)) may be of interest elsewhere.

Section (0) is devoted to preliminaries and notations to be used throughout. In Section 1 we state our results. The main results are Theorems (1.2), and (1.7). Proof of Theorem (1.2) is postponed to Section 2, but Theorem (1.7) is proved in this section itself. Section 2 is fully devoted to the proof of Theorem (1.2).

SHRAWANKUMAR

0. PRELIMINARIES AND NOTATIONS

(0.1) k, throughout, will denote a field of characteristic 0. All the vector spaces are considered over k. For a vector space V, V* will denote Hom,( V, k). For a Lie-algebra g, its universal enveloping algebra, as usual, will be denoted by U(g). Let M be a left U(g)-module; by M’ we would mean the right U(g)-module with the underlying space being the same as M and the action being m . a = 7’(a). m, for m E M and a E U(g), where T is the unique anti-automorphism of U(g) which is - 1 on g. Modules will be left unless explicitly stated. H,(g, M) would mean Torf(Q)(k, M) z Tor,U(g)(M’, k), for any n > 0.

(0.2) A symmetrizable generalized Cartan matrix A = (aG), Gr,jGl is a matrix of integers satisfying aii = 2 for all i, aV d 0 if i #j and DA is sym- metric for some diagonal matrix D = diag. (ql ,..., qr) with qi > 0 E Q.

Choose a triple (h, rr, rc”), unique up to isomorphism, where h is a vector space over k of dimension I+ co-rank A, rc = { ai}, <,</c h* and 7Cu= {hi)l<i<lCb are linearly independent indexed sets satisfying cr,(hj) = aii. The Kuc-Moody Algebra g= g(A) is the Lie-algebra over k, generated by h and the symbols e, and fi (1 d i< I) with the detining relations [h, h] =O; [h, ei] = Qh) e;, [h,fi] = -crj(h)fi for heh and all 1 <i<l; [ei,f;] =6,hj for all 1 <i,j<l; (ad e,)l-+(ej)=O= (adfJ-“l](J) for all 1 < i #j 6 1.

h is canonically embedded in g. Also, any ideal of g that intersects h trivially is zero itself by [GK].

(0.3) Root space decomposition [K4]. There is available the root space decomposition g = h @ C, Ed c h* ga, where grr = {x E g: [h, x] = cl(h) x, for all hE$} and d = {GLE~*\{O} such that ga ZO}. Moreover, A = A + u A ~, where A+ c {cf= 1 n,cl,: PZ~E Z + (= the non-negative integers) for all i} and A_= -A+. Elements of A + (resp. A _ ) are called positive (resp. negative) roots.

Define the following Lie-subalgebras:

n= 1 CL; n- = 1 ga; b=h@n; b- =hQnp. aed, atA-

The Lie-algebra n (resp. n - ) is graded with respect to the grading 1x1 =Cn, for xEg, (resp. xEg_,), where @=cf=, njcci. This gives rise to a grading on U(n) (resp. U(n)). We denote by U,(n) (resp. U,(n)) the homogeneous elements of U(n) (resp. U(n- )) of grade degree d.

(0.4) Symmetric bilinear form on IJ* [K4]. Define g(ai, aj) =qiuii, for

HOMOLOGYOFKAC-MOODY ALGEBRAS 447

1~ i,j < 1. Set h,, = q,h,. It is possible to extend (T to a nondegenerate sym- metric bilinear form, again denoted by G, on h* satisfying

a(A ai) = i(h,), for all 1 < i 6 I and all 1 E h*.

We fix one such 0. Identifying h with h*, this form extends to a non- degenerate invariant form on g.

(0.5) The category 0 denotes the full category of all the left g-modules A4, which are h-diagonalizable with finite dimensional weight spaces and whose set of weights lies in a finite union of sets of the form

( i

I D(v) = v- c n,fxi, rqEZ+

i) ) for v~h*.

i=l

(0.6) The Casimir operator [K2, K43. For each positive root c(, choose a basis {e;} of the space gcr and let {eir} be the dual basis of gPol. Define an operator, called the Casimir operator,

where v: h + h* is the canonical identification, {u”} and { uk} are any dual bases of h, and p is any fixed element of h* satisfying p(h,) = 1 for all 16iGZ.

52 acts on every module of the category 0. In fact, Q is a natural trans- formation of the category 0. Further, Q acts as scalar multiplication by [o(n + p, J. + p) - o(p, p)] on any highest weight module P’(n) with highest weight 1.

1. STATEMENT OF THE MAIN RESULTS

(1.1) Description of the algebra o(g) [CI]. Let g=g(A) be the Kac-Moody Lie-algebra, associated to a symmetrizable generalized Cartan matrix A. We briefly recall the definition of an associative algebra o(g), which is in some sense a completion of U(g), introduced by Vyjayanthi Chari and S. Ilangovan [CI].

Define o(g)= ndao (U(bh) OkU,(n)). For the description of the product structure in U(g), see [CI]. The algebra U(g) sits inside o(g) as c d, 0 U(b ~ ) Qk U,(n). (For various notations, see the previous section.)

Any left U(g)-module M in the category 0 admits a left o(g)-module structure, extending the U(g)-module structure on M. Further the Casimir operator D (described in Section (0.6)) can (and will) be viewed as an

448 SHRAWANKUMAR

element (again denoted by) Q of o(g) in the sense that the action of the Casimir operator Q on any module ii4 in the category Co is the same as multiplication by the element Q (E o(g)) in the extended o(g)-module M. Moreover, it can be seen that 0 is central in o(g).

Now we are in position to state one of our main theorems.

(1.2) THEOREM. Let g = g(A) be the Kac-Moody Lie-algebra associated to a symmetrizable generalized Cartan matrix A and let o(g) be the algebra described above. Let M be a left o(g)-module such that the element D E o(g) acts as an automorphism on M then

H,(g, M) = 0 for all i 3 0.

(M is viewed as a g-module by restriction.)

We prove this theorem in the next section. Also, we have the following simple proposition.

(1.3) PROPOSITION. Let g be as in Theorem (1.2) and let V(1) be any highest weight g-module with highest weight 2. Further assume that 2 4 Cf=, Zcc,, then again we have

ffi(S, VA)) = 0 for all i 3 0.

ProoJ Consider the standard chain complex {~I,=/i~(g, V(A))}iao (to compute the homology H(g, V(n))), h w ere /i,(g, V(L))=ni(g) Ok V(L) and d:Ai+Aj~, is given by

d(v, A ... A v,Oa)

=,<p~y<,(-l)p+4[VprVq]AV,A ‘.’ A8,A ... Ati,A “. AU,@U

. .

+ c (-l)%, A ... A tip A .” A U;@(U;U),

p=l for vi ,..., V,E g and a E V(2).

Clearly d is a g-module (in particular an h-module) map under the canonical g-module structure on /iIs. It is well known (and easy to prove) that g (and hence h) acts trivially on H,(g, V(L)). Further, /ii being weight modules, we have H,(g, V(L)) ~H,([LI]~) where [,41h denotes the sub- complex {/ip)i,O consisting of h-invariant elements in fti.

Weights of /li are of the form J + cf= i Zcl,. Since, by assumption, 2 $ cf=, Zai, the zero weight space of ni is equal to 0. This proves the proposition.

HOMOLOGYOFKAC-MOODY ALGEBRAS 449

Combining Theorem (1.2) and the above proposition, we get the following.

(1.4) COROLLARY. Let M be any g-module in the category 0. As is known [GL; Lemma 4.41, there exists a g-module filtration 0 = M, c M, c M, c . , such that M = Uj M, and each g-module M, + ,/M, (j>,O) is a highest weight module with highest weight Ai. Assume that, fbr any j>,O, either a(/Z,+p, 3L/+p)-o(p,p)#O or J,$Cf_IZai. Then Hi( g, M) = 0 for all i 3 0.

Proof Since the Casimir Q acts on M,, ,/M, as scalar multiplication by a(Ai + p, ,I, + p) - a(p, p), successively using the long exact sequence (see, e.g., [H; Sect. 23) corresponding to the coefficient sequence

we get that H,(g, Mj) =0 for all j. Since the functor Tor commutes with direct limits, we get the corollary.

We compute the homology of g with coefficients in arbitrary Verma modules and also the integrable highest weight modules. We have the following.

( 1.5) PROPOSITION. Let M(2) be the Verma module corresponding to any highest weight jl~b*. Then:

(a) If i # wp - p for any w E W (the Weyl group associated to g), we have

Hi(g, M(J-1) = 0 .for all i>,O.

(b) Zf l=w,p-p for some WOE W, then

Hi(g, M(A)) E Ai~‘(w”‘(b)

(l(w,) denotes the length of wO). In particular, Hi(g, M(1)) = 0 if i < l(w,).

(1.6) Remark. This result (stated in cohomology) in the case when g is a finite dimensional semi-simple Lie-algebra is due to Williams [W; Theorem 4.151. Our proof is very similar to his.

Proof. Let k, = k be the left b-module so that n acts trivially on kj. and h acts by the weight A. By the definition, M(I”) = U(g) 0 UCbI k,. Now by [CE; Proposition 4.2, p. 2751, we have (observe that there is a slight dif- ference between our notation and the notation used in [CE]. We are denoting Tory(“)(k, M) by H,(g, M), whereas it is denoted by H,(g, M’) in C-1.)

H,(b, k,) = ff,(g, t-4 Ouch, Vg)l’) (*I

450 SHRAWAN KUMAR

(Recall that for a left module M, M’ is defined in Section (0.1) and similarly we can define a left module M’ if we start with a right module M.)

But Ckj, @U(b) u(dl’ can be easily seen to be isomorphic with the Verma module M(1). So, from (*), we have

The Hochschild-Serre spectral sequence for homology (see [GE; p. 351 I), applied to the pair (b, n), has E& = H,(h, H&n, k,)) and it con- verges to H,(b, kn). Since n acts trivially on k,, we get that H&n, k,) is isomorphic with H&n, k) @ k, as h-modules. By [GL; Theorem 8.61, we have

ff,(n, k) = c k,- wpr WE Wwth

ew)=y

as h-modules.

Further, since h acts reductively on H&n, k) Ok k,, we have fWi f&h k) Ok kJ = ff,(h Cff& 4 OkkLlb) (C&h 4 OkUh denotes lj-invariants). This, in particular, gives that in the case (a), E& = 0 for all p and q, and in the case (b), [H,(n, k) OkkJh = 0 for q # I(w,) and Cf&,,(n~ W OkUh is one dimensional. So Eg,, = 0 for q # Z(w,) and E;,,(,,,, = ffp0-i k) = Nb).

In particular, the spectral sequence degenerates at the E* term itself, i.e., E* z E”. This proves the proposition.

Finally, we have the following generalization, of a theorem of Whitehead [J; Theorem 14, p. 961, to arbitrary symmetrizable Kac-Moody Lie- algebras.

( 1.7) THEOREM.’ Let L(A) be an integrable highest weight module (also called standard or quasi-simple module) with highest weight il #O (A is automatically dominant integral). Then, we have

Hi(g, L(A))=0 for all i20.

Proof: We recall the BGG resolution given in the general case by Garland-Lepowsky [GL; Theorem 8.71. There is an exact sequence of U(g)-modules:

. . . + E, + E, + E, -+ L(I) + 0,

’ This result, as well as the vanishing of H*(g, L(1)), has been obtained earlier by M. Duflo (unpublished, private communication to Victor Kac) by using J. Lepowsky’s Theorem (6.6) in Generalized Verma modules, loop space cohomology and Macdonald-type identities, Ann. Sci. l?ccole Norm. Sup. 12 (1979), 169-234.

HOMOLOGY OF KAC-MOODY ALGEBRAS

where each E, has a filtration

451

such that, for all 0 d id n(j) - 1, Ej+ ‘/Ej is a Verma module M($) with highest weight 1;. Moreover, up to a rearrangement, the set {A;>,, iGnop i coincides with { w(A + p) - p},+,, Wwith ,(w)=j. By Proposition (1.5), H,(g, M(w(l+p)-p))=O for any WE W, since w(Iz+p)-p is not of the form w,p - p for any WOE W. (Otherwise 0 #A = w-‘w,p -p. But w~lwOp-p= -cf=,niai with nie7+, which is not possible in view of [GL, Proposition 2.121.)

By the same argument used in the proof of Corollary (1.4), we get H,(g, E,) = 0 for all j> 0. But then H,(g, L(1)) z Tori’“‘(k, L(I)) is the homology of the complex (see the proof of Lemma 2.3))

But k @ u(e) E,= Ej/U+(g).Ej=O (since clearly M(A)/U+(g).M(A)=O for A # 0).

Hence the theorem follows.

(1.8) Remarks. (a) The above theorem can also be deduced immediately from our Theorem (1.2) for those L(A) such that c(A+ p, A+ p) - o(p, p) # 0. In the case when g is finite dimensional, afline, or hyperbolic, it is indeed true that a(l+p, ;1 +p)- o(p, p) #O for all dominant integral elements A #O. (See [K3; Sect. 11.) But in general (as Victor Kac has pointed out) it is not true. The following example is due to him. Take

2-l 0 0

-1 2 -6 0 A=

i 1

0 -6 ’ 2 -1

0 o-1 2

then 3p =a1 -~~---~+a, and o(p, p)=O. So, if we take p=kkp (k>O), then 4~ + P, P + P) - +, P) = 0.

(b) We prove [Ku; Theorem 1.61 that H,([g, g], k) ([g, g] denotes the commutator subalgebra of g) is isomorphic with the singular homology of the corresponding algebraic group G (or its subgroup K). See also [KS]. (G and K are defined in [KP].)

We come to the computation of H,(g, k). Consider the canonical surjec- tion 7~: g + g/[g, g]. From the root space decomposition, rc(h) = g/[g, g].

452 SHRAWANKUMAR

Hence there is a Lie-algebra splitting s: g/[g, g] + g (i.e., s is a Lie-algebra homomorphism and rt 0 s = Id.). Fix any A E h * such that A 1 h n rs, 91 = 0. We define a l-dimensional g-module kA by demanding that [g, g] acts trivially on kl and h acts by the weight A on kl. From an analogue (though proved in cohomology and under the assumption that g is finite dimensional, it is not difficult to see that their proof can be easily adopted to our situation) of [HS; Theorem 133, we get

i.e., (since g/[g, g] is abelian)

Next we prove that H,( [g, g], k) is trivial as a g-module. For, if not, let kj, (A # 0) be a direct summand of H,( [g, g], k) as an h-module. (Since H,([g, g], k) is trivial as a [g, g]-module, 1 satisfies that Alhn c9,9, =O.) From (**), we get

H,h k-d=NglCg, d)O CH,(Cg, rrl, k-,)1”

=4glCg, gl)O Cff,(Cg, gl, k)Ok-2 (since H,(Cg, 91, k-J ~:~([g,g],k)@k_,asg-modules)

#O (since, by assumption, kn is a direct summand of H,(Cg, 91, k)).

But by Theorem (1.7) H,(g, kp,) = 0 (hie [g, g], so A(hi) = 0 and hence -A is dominant integral). This contradiction proves that H,( [g, g], k) is trivial as a g-module as well. From (**) (taking A= 0), we get the following

(1.9) PROPOSITION. H,(g, k)=:/i(g/Cg, g1)0H,(Cg, 91, k).

(1.10) Conjecture. Let AE~* be an element such that A # wp --p for any w E IV. It seems reasonable to conjecture that Hi(g, I’(A)) =0 for all i > 0 and for any highest weight module V(A) with highest weight 1.

This conjecture is trivially true in the case when g is finite dimensional, because the centre of U(g) separates the Weyl group orbits.

2. PROOF OF THEOREM (1.2)

The main ingredient in the proof of this theorem is the following proposition.

HOMOLOGYOFKAC-MOODY ALGEBRAS 453

(2.1) PROPOSITION. H,(g, l?(g))=Ofor all pb 1, where g acts on o(g) as the left multiplicution.

We postpone the proof of this proposition and first use it to derive our main theorem (1.2). We have the following lemmas.

(2.2) LEMMA. The canonical map

is an isomorphism.

Proof: Let &d+ =(W-))+OFL,.,[W-1 Ok Uc,(n)l, where (U(b ))’ denotes the two-sided ideal generated by b- in U(bh). It s&ices to prove that any element a in U(g)+ can be written as a = x7=, X,ai, for some Xi E g and uie U(g). This follows essentially from the fact that the algebras U(b - ) and U(n) are both finitely generated. 1

Write a E U(g)+ as CdaO ud with USE U(bb) Ok U,(n). Further, any ad can be written as

with h:, and &;E U(bh)@ U,(n) for all i and j and CUE Udp ,(n) (ei,fi are defined in section (0.2) and 4. is any k-basis of h).

Summing over d, this gives the lemma.

(2.3) LEMMA. Tory’g)(k, M) z Torp(“)(k, M), for any left C?(g)-module M and for all i.

Proof Let us take a U(g)-free resolution of the U(g)-module M

+ F2 + F, + F, + M -+ 0.

From Proposition (2.1), Toq%)(k, F,) = 0 for all i > 0 and all j 3 0, since Hi(g, U(g)) is defined to be Toru(fl)(k, U(g)). So TorU(“) (k, M) is given by (see [G]) the homology of the complex

Of course, Tor”‘“‘)(k, M) is given by the homology of the complex

We have the following commutative diagram:

454 SHRAWAN KUMAR

From Lemma (2.2), all the vertical maps are isomorphisms. This proves the lemma.

(2.4) Proof of Theorem (1.2). Since Hi(g, M) is defined to be Tory(g)(k, M), in view of Lemma (2.3), it s&ices to show that Toru(g)(k, M) = 0. But the Casimir Q is a central element in the algebra o(g), which acts as 0 on k and as an automorphism on A4 (by assumption). From this, the vanishing of Tor,?(“)(k, M) follows easily. 1

We come to the proof of Proposition (2.1). Let us recall the following well known facts, which will be used often.

(1) ffi(a, a’, M) * z H’(a, a’, M*), for any Lie-algebra a and a sub- algebra a’ with A4 being a left a-module.

(2) The map e: Hom,( V, W*) + [I’ Ok IV]* defined by e(f)(u@w)=f(u)(w), for fEHom,(V, IV*), VEV and WE W, is an a- module isomorphism for any left a-modules I’ and W.

(2.5) LEMMA. HJg, fi(g))zHJg, bb, ir(g)),for aZZp>O.

Proof The Hochschild-Serre spectral sequence [HS; Sect. 21, applied to the Lie-algebra pair (g, b-) with coefficients in 6(g)*, is a convergent spectral sequence with

EyY= HY(b-, Hom,(AP(g/b-), o(g)*)) z HY(b-, [/i”(n) 0,0(g)]*)

and converging to H*(g, o(g)*). In a subsequent lemma (Lemma (2.7)), we prove that

HY(b-, CAP(n) ok o(g)]*) =O for all q>O and all ~20. This gives that EpY = 0 for all r > 1 and q > 0. Hence Eqq z E%q for all p and q. Further E$O= HP(g, b-, I?(g)*). (See [HS; Sect. 21.) This gives that the canonical restriction map HP(g, bb, 8(g)*) + HP(g, o(g)*) is an isomorphism for all p. But then H,(g, o(g)) + H,(g, b-, o(g)) itself is an isomorphism. (Use the fact that if V and W are two k-vector spaces with a linear map f: V+ W such that the induced dual map f *: W* + V* is an isomorphism, then f itself is an isomorphism.)

(2.6) LEMMA. Hq(u, [/i”(n) Ok o(g)]*)=0 for all q>O and for all p 3 0, where n = g/b ~ is a U(n- )-module under the udjoint action and 6(g) is a U(n- )-module under the left multiplication.

HOMOLOGY OF KAC-MOODY ALGEBRAS 455

Proof: There exists a U(n ~ )-resolution of the trivial module k

such that all the Fq’s are free and finitely generated left U(n ~ )-modules. In fact, by [GL; Section 71, Fq can be further chosen to be a graded sub- module of U(n) Oknq(n-), such that the map dq is the restriction of the differential in the canonical resolution of the trivial module k:

... -+ U(n-) Ok/lq(n-) -+ .. . -+ U(n-) Ok/i’(n) -+ k.

(We take the product grading on U(n) Ok .4”(n).) The latter fact will be used only in a subsequent lemma (Lemma (2.9)).

Any u E &(n) is contained in a finite-dimensional (over k) U(n - ) - invariant submodule M, of rip(n). (This is easy to see in view of the fact that the root spaces gbl are finite dimensional.) Consider the finitely generated and free resolution of the U(n ~ ) - module M, :

‘.. -+Fq@kM, dq,o 1 ,F q-, OkMu-+ ... +F’,QkM,

doQ1 -f@% Fo’OOkMv- M,.

(The fact that Fq ok M,‘s are U(n- )-free, follows from the Hopf principle [GL; Proposition 1.71.)

Let K, = ker(d, 0 1 = aq). We have the exact sequence

F q+zOkMu “+’ ,Fq+, OkM, “+’ b KpO,

and hence K, is finitely presented as a U(n ~ )-module. We prove that

is injective for all q > 0. Consider the following commutative diagram:

[, dlJo[U(b-) &Ud(n)l 1 ~~,,.-,Kq=d~o[[U(b)Q*Lidn)lr~~~~~rKql

I 1 Qg)’ 0 ucn-j(Fq QkKb n WV-) Ok Ud(n)l'@u(n-)(Fq GhM,)l. d>O

456 SHRAWAN KUMAR

The top horizontal map is an isomorphism. (See [B; Sect. 3, No. 71.) The right vertical map is injective, as [ U(b - ) Ok UJn)] ’ is free right U(n- )-module. So the left vertical map is also injective. This gives (on using the right exactness of 0) that Tory’“-)( o(g)‘, M,) = 0 for all q > 0. Which, in turn (since AP(tt) is a direct limit of M,‘s) implies that

Tory(“-)( o(g)‘, k’(n)) z Tor~(“~)(k, o(g) Ok /i”(n))

= H,(n-, O(g) Q/&P(n)) = 0

for all q > 0. This proves the lemma. Now we come to the following

(2.7) LEMMA, HY(b -, [A”(n) Ok o(g)]*) = Ofor all q > 0 and UN p 20.

Proof: Consider the Hochschild-Serre spectral sequence relative to an ideal, corresponding to the pair (b-, n) (coefficients being CA”(n) Ok h)l*L with

Ep’= fP(l~, HS(nP, [P(n) Ok %)I*))

and converging to H”+“(b-, [P’(n) Ok @g)]*). From Lemma (2.6), w(n.-, CAP(n) ok o(g)]*) = 0 for all s > 0 and hence the spectral sequence degenerates at the E, term itself. So Hq(bP, CAP(n) Ok p(g)]*) z Hq(h, @(n-, [A”(n) Ok o(g)]*)).But @(n-, [/l”(n) OkU(g)]*) is the space of invariants in [A”(n) Ok o(g)]* which is equal to CAP(n) Ok @3)/n. CAP(n) Ok~kdll *. Now from an argument com- pletely analogous to the one used in the proof of the next Lemma (2.8), we get that this is isomorphic (as an h-module) with

Hence

Ap(n) Ok n [u(b) OJUn)l *. d>O 1

HY(b-, [A”(n) Ok o(g)]*) z Hq A”(n) Ok n [U(h) Ok Ud(n)] * d I>

z Tor,U(“) (P(n)‘, n [U(h) ok U,(n)])*. d20

HOMOLOGY OF KAC-MOODY ALGEBRAS 457

But U(h) being a commutative Noetherian ring, ndaO [U(h) Ok Ud(n)] is a flat U(h)-module [C; Theorem 2.11 and hence the lemma follows.

(2.8) LEMMA. 11,(g, b-, o(g))% H,(n, nda,, U,(n)) for all p, where rI da0 U,(n) is considered as a left U(n)-module by left multiplication.

Proof. Let A(g,b-, U(g))=(... -+A,+d~AP_I-+ ... -+A,-+O) be the standard complex for computing the homology H(g, b -, U(g)). We recall that

and

A,, = CAP(n) Ok &dl/bp . CAP(n) Ok &x)1

= 1 (-l)i+.i[U;, U,] A VI A “’ A tii A “’ A fij A .‘. A U,,@U

I<,<j<p

+ i (-l)'U, A ... A tii A ... A U,@(Ui.U),

i= I

for u1 ,..., u,, E n and a E U(g),

where u, A ‘.. A up@ a denotes the projection of u1 A “. A up@ a E

P(n) Ok CT(g) onto AP. We define a chain map p= (BP) with BP: A,(& nd U,(n)) -+

A,(g, b-, U(g)) as follows:

foruEP(n) and xdEUd(n).

Further we define a map a,(d): AP(n) OkU(bP) OkUd(n)+ /i”(n) Ok U,(n) by a,(d)(u@b@x,)= T(b). u@xdfor uE/IP(n), bE U(bb), and xd E u,(n). (Recall from Section (0.1) that T is the unique anti- automorphism of U(bb) such that T(X) = -X for XE b-.) This gives rise to a map

a = up: A”(n) ok n CUW) ok Ud(n)l + A”(n) ok n ud(n),

da0 da0

defined by u 0 (C d>Obd@Xd)H~dT(bd).U@XdfOrUE/iP(lt), bdEU(b-), and xd E U,(n). This map is well defined, since U(b - ) . u is finite-dimen- sional subspace of AP(n) for any ueAP(n) (see the proof of Lemma (2.6)).

458 SHRAWAN KUMAR

The map cx can be easily seen to factor through b- . [A”(n) Ok 6(g)]. We denote by a the map 0: defined on the factor space

A”(n) Q,c @9)/b- . C4n) Ok &dl.

Clearly ZoB=Id. Further, to prove that (PoCI)x-XE b- . [A”(n) Ok U(g)] for all x E AP(tt) ok U(g), observe the following:

(1) uOXu= -(Xu)@amodbb. [U(bb).u@,U(bb)] for am&‘, XE b-, and uE U(bb),

(2) U(b -). v is finite dimensional /k, for all u E Ap(n), (3) U(b- ) is a finitely generated algebra /k.

This proves that the map a is an isomorphism of the chain complexes and hence the lemma follows.

Finally, we have the following

(2.9) LEMMA.’ H,(n, ndso U,(n)) =Ofor all p 2 1.

Proof Choose a U(n)-resolution of the trivial module k (as stated in the proof of Lemma (2.6)):

with F, a graded submodule of U(n) Qk P(n). Further F, is free and finitely generated left U(n)-module and all the differentials d, are graded morphisms.

Let K, = ker d,,, then K, is finitely generated graded U(n)-module with homogeneous generators (say) {u,,..., o,}. For a U(n)-module M, let fi denote its m-adic completion with respect to the ideal m = U(n)’ of U(n). We prove that & + & is injective for all p 2 0. For this, it suffices to show that for any n, there exists n’ such that

[m”’ * F,] n Kp c m” * K,.

Let x E Cm”‘. F,] n K, be a homogeneous element of degree d, so that d 2 n’. Of course x can be written as x = I;= i aiui with ui E U(n). Further, we can choose ui to be homogeneous of degree (d- deg. ui). So, if we take

2 M. V. Nori has pointed out the following very short proof of this lemma: The sequence + I, + ~~ _ ,(d) -P + F,,(d) -+ k(d) -+ 0 is exact, where F,(d) denotes the elements of

grade degree d in FP, and hence + ndzo F,(d) +&F,-,(d) + -+ l-Ida0 F,(d) -P k + 0 is also exact. But ndro p F (d) can be canonically identified with [rJd>O ClAn)]’ C3”cn, FP.

HOMOLOGY OF KAC-MOODY ALGEBRAS 459

n’ = max IciG,{degui+n} then degaian. Hence aiEm”, as m”= Ci,,, U,(n). (This follows easily from the fact that U(n) is generated, as an algebra, by its elements of degree 1.) This shows that x E m” . I$, so the assertion that gp injects into flp is established.

Now we want to show that [-Ida0 U,(n)lr BuCn) K, injects into [IX dZO Udn)l Ouctlj F, for all p 2 0, which will prove the lemma.

The k-algebra ndZo U,(n) can be identified with the k-aigebra C?(n), where o(n) is the m-adic completion of the k-algebra U(n) with respect to the two-sided ideal m. There exists a canonical map C?(n)’ 0 uCn) K, + kp, induced from the map [ U( n)/md] ’ @ UC ,, ) K, + Kp/md. K, defined by G@XH r(a). x for UE U(n) and XE Kp. We have the following com- mutative diagram:

I 1 I

ppp+2- F P+l + kp+O.

The top horizontal map is exact due to right exactness of 0. The first two left side vertical maps are isomorphisms, because F, + 2 and F, + , are free and finitely generated. To prove the exactness of the bottom sequence, we have the exact sequence:

and hence the following sequence is also exact (see [M; Proposition on p. 1671).

o-+Icp+, -+Fp+l 41p+o.

Similarly we have the exact sequence:

O’kpt2’~ppt2’Icp+l’0.

By (S,) and (S,), we get that

(S,)

c%)

is exact. So, using the five lemma, we get that o(n)’ 0 vCn) Kp % &. This establishes the lemma.

(2.10) Proof of Proposition (2.1). Lemmas (2.5), (2.8), and (2.9) put together immediately give Proposition (2.1).

460 SHRAWAN KUMAR

(2.11) Remark. Suppose we take a larger completion o*(g) = &,JU(n-) @%@ UJn)] instead of o(g), where % =I S(h) is an ‘appropriate’ class of functions on a subspace of b*, and ask if our “crucial” (to the proof of Theorem (1.2)) Proposition (2.1) and Lemma (2.2) are true with o(g) replaced by OF(g). This would enable one, with an appropriate choice of %, to get many more central operators (other than the Casimir). (See [K5].) This, in turn, could be used to prove our vanishing theorem ( 1.2) for a larger class of modules M. See also conjecture (1.10).

But we briefly sketch an argument to show that if % I> S(b) satisfies the following:

(a) % is a commutative ring with an augmentation E: % -+ k (i.e., E is a ring homomorphism such that ~0 i= Id., where i: k 4 S(t)) is the canonical inclusion) such that s(t)) = 0;

(b) % is free as a S(h)-module; (c) h generates % + = ker E as %-module.

Then % = S(h).

(Observe that our proof of Proposition (2.1) and Lemma (2.2) uses the properties (a), (b), and (c) above.)

Take a S(h)-free basis {en}nt, of %. By assumption

e+ = e, - i&(e,) = c 1 h, PJfm e,, n for some PIm E S(f)), (*) me1 1’

where {hj} is a k-basis of b and P&, = 0 for all but finitely many m. Write 1 =CmEIfmemT with f, E S(h) and all but finitely many J,, = 0. Hence, by (*),

hj Pjl,,, + de,) fm

SO, CjhjP&+&(en)fm=6n,m for all n, m E I. This, in particular, shows that Z is finite and hence % is a Noetherian ring. (Otherwise we can choose n, E Z such that f, = 0. So cj hi PITo = 1 in S(h), a contradiction!)

By induction, it is easy to see that % = S’(b)“. 9 + S(b) for all p 2 1 (use %+ c (%+)‘+ S+(b)). In particular the S(h)-module %/S(b) satisfies

S’(b)“. (%/S(h)) =%/S(b), so the completion %/S(h) of the S(h)-module

%/S(h) along S+ (5) is 0. This gives that 3 ~3 %. But 9 being S(b)-free, rank % = 1. Now, it is easy to see that % = S(h).

HOMOLOGY OF KAC-MOODY ALGEBRAS 461

ACKNOWLEDGMENTS

My most sincere thanks are due to Vyjayanthi Chari, with whom I had many discussions and who declined to be a joint author of this paper. I am grateful to Victor Kac for his many valuable comments, which helped to bring this paper to its present form. My thanks are also due to R. C. Cowsik, M. V. Nori, Gopal Prasad, and D. N. Verma for some helpful conver- sations.

Note added in prooJ:

(1) This paper was earlier circulated as MSRI preprint No. 033-84-7 with the title “Homology of Kac-Moody Lie-algebras with arbitrary coefficients”.

(2) We prove a fairly general vanishing theorem for the Ext bi-functor (which, in par- ticular, gives our theorem (1.2)) by a different method, in “Extension of the category 08 and a vanishing theorem for the Ext bi-functor for Kac-Moody algebras” (to appear in J. Algebra).

(3) In the above paper, we also settle conjecture (1.10) in affirmative for the affine (including twisted afline) algebras. For general symmetrizable Kac-Moody algebras, we show that if A E K”-r- (it is defined in the above mentioned paper and it contains the Tits cone), the conjecture is again true.

(4) We learned of a paper by Chiu Sen “The homology of KaccMoody Lie algebras with coefficients in a generalized Verma module,” J. Algebra 90 (19&l), where Proposition (1.5) is proved.

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462 SHRAWAN KUMAR

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